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Universidade de Vigo
A nonparametric test for markovianity
in the illness-death model
Mar Rodríguez Girondo and Jacobo de Uña Álvarez
Report 11/04
Discussion Papers in Statistics and Operation Research
Departamento de Estatística e Investigación Operativa
Facultade de Ciencias Económicas e Empresariales Lagoas-Marcosende, s/n · 36310 Vigo
Tfno.: +34 986 812440 - Fax: +34 986 812401 http://webs.uvigo.es/depc05/
E-mail: [email protected]
Universidade de Vigo
A nonparametric test for markovianity
in the illness-death model
Mar Rodríguez Girondo and Jacobo de Uña Álvarez
Report 11/04
Discussion Papers in Statistics and Operation Research
Imprime: GAMESAL
Edita: Universidade de Vigo Facultade de CC. Económicas e Empresariales Departamento de Estatística e Investigación Operativa As Lagoas Marcosende, s/n 36310 Vigo Tfno.: +34 986 812440
I.S.S.N: 1888-5756
Depósito Legal: VG 1402-2007
A nonparametric test for markovianity in theillness-death model
Mar Rodrıguez-Girondo1 and Jacobo de Una-Alvarez1,2
1SiDOR Research Group, University of Vigo. Address: Facultade de CC Economicas
e Empresariais, Campus Lagoas-Marcosende, 36310, Vigo, Spain. E-mail: mar-
[email protected] of Statistics and OR, University of Vigo. Address: Facultade
de CC Economicas e Empresariais, Campus Lagoas-Marcosende, 36310, Vigo,
Spain. E-mail: [email protected]
Abstract
Multi-state models are useful tools for modeling disease progression when
survival is the main outcome but several intermediate events of interest are
observed during the follow-up time. The illness-death model is a special
multi-state model with important applications in the biomedical literature. It
provides a suitable representation of the individual’s history when an unique
intermediate event can be experienced before the main event of interest.
Nonparametric estimation of transition probabilities in this and other multi-state
models is usually performed through the Aalen-Johansen estimator under a
Markov assumption . The Markov assumption claims that given the present state,
the future evolution of the illness is independent of the states previously visited
and the transition times among them. However, this assumption fails in some
applications, leading to inconsistent estimates. In this paper we provide a new
approach for testing markovianity in the illness-death model. The new method is
based on measuring the future-past association along time. This results in a deep
inspection of the process which often reveals a non-markovian behaviour with
1
different trends in the association measure. A test of significance for zero
future-past association at each time point is introduced, and a significance trace
is proposed accordingly. Besides, we propose a global test for markovianity
based on a supremum-type test statistic. The finite sample performance of the
test is investigated through simulations. We illustrate the new method through
the analysis of two biomedical data analysis.
Keywords Multi-state models; Illness-death model, Markov condition; Kendall’s
Tau.
1 Introduction
Multi-state models [1, 2, 3] are typically used for modeling disease progression
when several intermediate events of interest are observed during the follow-up
time. These models are an extension of the traditional survival analysis and make
it possible to account for complex individuals’ history with a possible influence on
the prognosis. In particular, the traditional survival analysis refers to the simplest
multi-state model, the mortality model, where only two states are considered, an
initial (‘alive’) state and a final absorbing state (‘dead’).
Mathematically, a multi-state model refers to a stochastic process varying in
continuous time{X(t), t ≥ 0}, whereX(t) is the state of the individual at time
t and allowing individuals to move along a finite number of states. Furthermore,
we assume that the trajectories of individuals can be right censored by a potential
censoring time that is independent of the process. In biomedical applications, the
states might be based on clinical symptoms (e.g. bleeding episodes), biological
markers (e.g. CD4 T-lymphocyte cell counts), some scale of the disease (e.g.
stages of cancer or HIV infection), or a non-fatal complication in the course of the
illness (e.g. cancer recurrence).
In this work, we focus on the illness-death model, which involves three differ-
ent states by splitting the ‘alive’ state in two different states (1=‘healthy’, 2=‘dis-
2
eased’) and considering a last absorbing state (3=‘dead’) and three possible tran-
sitions among them1→2, 2→3 and1→3 (see Figure 1). Many applications of
the illness-death model can be found in the biomedical literature [4, 5, 6, 7, 8].
In section 4, we reanalyze two biomedical datasets in a new scope. Firstly, we
propose to consider an illness-death model for analyzing the influence of the time
to chronic graft-versus-disease in evolution after a bone marrow transplant in pa-
tients with leukemia [9]. Secondly, we consider the relationship between the time
to the first wound excision and time to Straphylocous Aureaus infection in burn
patients [10]. It is noteworthy that the illness-death model contains other simpler
schemes, such as the mortality model and the three-state progressive model where
direct transitions between state 1 and state 3 are not possible.
1.Healthy 2.Diseased//
3.Dead��
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Figure 1: Illness-death model
Denoting byZ the sojourn time in state 1 and byT the total survival time, the
illness-death model is characterized by the joint distribution of (Z, T ). Note that
individuals who transit directly from state 1 to state 3 are those withZ = T , while
Z < T indicates that the individual has passed through the intermediate state 2.
The Markov assumption claims that given the present state, the future evolution of
the illness is independent of the states previously visited and the transition times
among them. This condition is sometimes violated. For example, [6] provide evi-
dence on non-Markovianity in a study on mortality in liver cirrhosis; in this case,
the Markov condition fails because the mortality is markedly increased shortly
after undergoing a bleeding episode, which is taken as an intermediate event. One
of the main arguments for checking the Markov condition is that the usual estima-
3
tors (e.g. Aalen-Johansen transition probabilities) may besystematically biased in
non-Markovian situations [11]. Despite non-Markovian estimators for transition
probabilities and related curves are available [11, 12, 13], it has been quoted that
including the Markov information in the construction of the estimators allows for
variance reduction [11]. So, in practice, one will be interested in testing marko-
vianity.
Letλs(.) denote the hazard function ofT for those individuals going from state
1 to 2 exactly at times (that is, for the subpopulation{Z = s, Z < T}). Note that
for t ≥ s, the hazardλs(t) is the instantaneous probability of ‘death’ for those
individuals being in state 2 at time t, which may be influenced by the specific
transition times. The Markov assumption states thatλs(t) does not depend on
the particular value ofs, t ≥ s. Traditionally, the Markov assumption is checked
by testingH0 : β = 0 under the proportional hazard model [4]λs(t) = λ0(t)eβs,
t ≥ s. However, the proportional hazard specification may not properly represent
the dependence structure between the survival prognosis and the sojourn time in
state 1 since both the linearity and proportional hazards may fail in practice. As a
consequence, this approach may be unable to detect the lack of markovianity. See
also sections 3 and 4. Therefore, alternative, more flexible methods are required.
Alternatively, the Markov condition can be formulated as the independence
between variablesZ andT conditionally on the eventAt = {Z ≤ t < T} (‘being
in state 2 at time t’), for each givent > 0. If we denote byH0t : T ⊥ Z
givenAt, t > 0, where⊥ stands for the independence relationship, thenH0 =⋂
t H0t represents the Markov condition in the illness-death model framework.
The general idea of the new methods is based on determining -without assuming
any predefined structure- the grade of dependency between past (Z) and the future
(T ) given the present. For this, the Kendall’s tau will be used in a local way.
The rest of the paper is organized as follows. In Section 2 we introduce the
new goodness-of-fit method for testing markovianity in the illness-death model
based on measuring future-past association. We introduce a bootstrap resampling
4
plan to approximate the null distribution of the test statistic. In section 3 a sim-
ulation study shows the performance of the new methods. Also, in section 4 the
new methods are applied to two real biomedical datasets and compared to the tra-
ditional Cox proportional hazards approach. Main conclusions and a final discus-
sion follow in section 5, while the technical details are deferred to the Appendix.
2 Main results
2.1 Goodness-of-fit tests based on Kendall’s tau
Focusing on the illness-death model discussed in the Introduction, letZ andT be
the sojourn time in state 1 and the time to reach state 3, respectively. As indicated,
the Markov condition can be formulated as the conditional independence between
T (future) andZ (past) givenAt = {Z ≤ t < T} (being in state 2 at the ‘present
time’ t for eacht > 0).
We use as measure of association the Kendall’s Tau:
τt = pc,t − pd,t, t > 0,
wherepc,t andpd,t are the probability of concordance and discordance, respec-
tively, for two pairs(Z, T ) falling onAt; more explicitly,
pc,t = 2
∫ ∫Ft(x
−, y−)Ft(dx, dy), pd,t = 2
∫ ∫Ut(x, y)Ft(dx, dy),
whereFt(x, y) = P (Z ≤ x, T ≤ y|At) stands for the joint distribution function
(df) of (Z, T ) conditionally onAt, and whereUt(x, y) = P (Z > x, T < y|At) =
Ft(t, y−)− Ft(x, y
−).
We firstly propose a method to test the null hypothesisH0t : τt = 0 for each
fixed time point. The basic idea is to rejectH0t for large values of an estima-
tor of |τt|. Since the Markov condition is represented by the intersection null
H0 =⋂
t>0 H0t, evidence against any specificH0t can be interpreted as a lack of
5
markovianity of the process. On the other hand, since many local tests are per-
formed at the same time, one may ask about the increase of type I error rates due
to the multiplicity of local tests . As a solution, we propose a global test taking
supt |τt| as a summary measure.
In most practical cases, the sample will be subject to right-censoring because
follow-up time limitations, withdrawals, and so on. LetC be the censoring time,
which we assume to be independent of(Z, T ). Furthermore, letZ = min(Z,C)
and T = min(T, C) be the censored versions ofZ andT respectively, and let
∆1 = I (Z ≤ C) and∆ = I (T ≤ C) be their corresponding censoring indica-
tors. The sample information is represented by{(
Zi, Ti,∆1i,∆i
), i = 1, ..., n
},
iid copies of(Z, T ,∆1,∆
). In practice, only situations(∆1,∆) = (0, 0), (1, 0),
and(1, 1) may happen, corresponding to individuals censored in state 1, in state
2, or uncensored, respectively. Besides, we assume that the individuals observed
to pass through state 2 coincide to those withZ < T (in words: we exclude the
possibility of a zero transition time from state 2 to state 3).
Therefore, for censored samples, we can introduce an alternative association
measure,τt, which is defined as the Kendall’s tau between the censored versions
of T (T ) andZ (Z) given At ={Z ≤ t < T
}for eacht > 0; τt = pc,t − pd,t
where
pc,t = 2
∫ ∫Ft(x
−, y−)Ft(dx, dy), pd,t = 2
∫ ∫Ut(x, y)Ft(dx, dy),
where Ft(x, y) = P (Z ≤ x, T ≤ y|At) stands for the joint df of(Z, T
)
conditionally onAt, and whereUt(x, y) = Ft(t, y−) − Ft(x, y
−). Note that
At = At ∩ {C > t} refers to the subpopulation being in state 2 at timet, which
has not been censored by that time. As long asC is independent of the process,
this subpopulation is representative of the subpopulationAt. Put H0t : τt = 0.
Interestingly, whenT andZ are independent conditionally onAt we have that
Z and T are independent givenAt ={Z ≤ t < T
}. The converse is also true
6
provided that the support ofC contains that ofT . This is stated as a Theorem.
Theorem 1 If T and Z are independent conditionally on At, then T and Z are
independent conditionally on At. Conversely, if T and Z are independent condi-
tionally on At, and if the support of T is contained in that of C, then T and Z are
independent conditionally on At.
Proof. See the Appendix.
Theorem 1 is useful, because it ensures that ifτt 6= 0 for somet we may
conclude that the process is not Markovian. Since variablesT and Z in which
τt is based are completely observable, one may use ordinary estimators. This
avoids the use of Kaplan-Meier-based estimators, which typically exhibits a large
variance under heavy censoring, and hence they lead to tests with a small statistical
power. See [14] for more on this in the scope of the three-state progressive model.
However, one should not take the valueτt as the (local) future-past association
between the original variables of the process, sinceτt andτt will be different in
non Markovian situations (except for the uncensored case, in whichτt = τt). In
order to illustrate this, in Figure 2 we report the traces ofτt andτt for the non-
Markovian Model 1 in section 3, for several censoring degrees; we can appreciate
how both traces become more and more distinct as the censoring grows. Indeed,
this Figure indicates that much less power should be expected under heavy cen-
soring, because the trace of the Kendall’s tau is closer to zero. For the Markovian
Model 0 in section 3, however, both traces coincide (and they collapse to zero).
We defineτ t in the obvious way:τ t = pc,t − pd,t where
pc,t = 2
∫ ∫F t(x
−, y−)F t(dx, dy), pd,t = 2
∫ ∫U t(x, y)
F t(dx, dy),
where U t(x, y) =F t(t, y
−)− F t(x, y
−), and whereF t is the ordinary empirical
distribution function of thent pairs(Zi, Ti
)satisfyingZi ≤ t < Ti. Since, this is
just the ordinary Kendall’s tau computed from a subsample, results on consistency,
finite sample distribution, and asymptotic normality are well-known. We refer the
7
interested reader to [15] and [16]. Hence, critical points for the local test under
the null can be computed as usual.
Problems arise when considering the statistic for the global test, namelyDn =
supt
∣∣∣τ t∣∣∣. The asymptotic normality of the finite-dimensional distributions of the
process(τ t)t>0 follows from the standard results on the Kendall’s tau and the
multivariate Central Limit Theorem. Therefore, the weak convergence of(τ t)t
to a Gaussian process -and hence the existence of a limiting distribution forDn
-follows provided that the process is tight. This is stated by Theorem 2 below.
However, the computation of the critical points from the asymptotic distribution
of Dn is difficult, and therefore we propose in the following subsection a bootstrap
resampling plan.
Theorem 2.Let I be an interval such thatP (At) ≥ c > 0 for all t ∈ I. Then,
the process{√
n(τ t − τt) : t ∈ I}
weakly converges to a zero-mean Gaussian
process.
Proof: See the Appendix.
2.2 Approximation of null distribution. Bootstrap approaches
In this subsection we propose a bootstrap approximation to the null distribution
of both τ t andDn = supt
∣∣∣τ t∣∣∣. Although this is not needed for the local test
(due to the availability of tables with critical points and asymptotic theory), for
completeness we give a bootstrap proposal (termed as Local Bootstrap, LB) also
for this case. The LB draws independently theZ and theT at eachAt. Therefore,
it is specific for each particulart value. On the other hand, to deal with the global
test a Global Bootstrap (GB) resampling plan is needed. This is also given below.
Note that the GB may also be applied to the local test, in order to incorporate the
full Markov information in the (local) testing procedure.
Local Bootstrap (LB):
The local bootstrap resample plan proceeds in two steps:
8
Step 1: Draw Z∗i from the marginal distributionF t(.,∞).
Step 2: Draw independentlyT ∗i from the marginal distributionF t(∞, .).
In Step 1, eachZi corresponding to a pair(Zi, Ti) falling on At is sampled
(with replacement) with probability1/nt, wherent =∑n
j=1 I(Zj ≤ t < Tj).
In Step 2, the valueTi is sampled in a similar way. Note that, in this man-
ner, new combinations of type(Zi, Tj) may appear in the bootstrap resample.
For each fixed value oft, steps 1 and 2 are repeated until a bootstrap resample
{(Z∗i , T
∗i )}nt
i=1 of sizent is generated. This bootstrap approach has the advantage
of its simplicity and computational efficiency but it presents the drawback that it
does not permit to manage the process in a global way and hence it cannot be used
for the global test.
Global Bootstrap (GB):
The global bootstrap algorithm is based in the global relationship betweenZ and
T underH0 =⋂
t>0 H0t Note that, under the null hypothesis, the process is
Markov, and hence a markovian estimator ofFT /Z(t/x) = P (T ≤ t|Z = x, Z <
T ) should be used to resample the total survival time given the observed transition
time from 1 to 2 (Z = x).
Explicitly, we used the following resampling algorithm:
Step 1: Draw Z∗i from F 1
Z.
Step 2: GivenZ∗i , drawT ∗
i from FT /Z(.|Z∗i )
In Step 1F 1Z
stands for the ordinary empirical df ofZ givenZ < T .
In Step 2, we takeFT /Z(t|x) = 1− p22(x, t) where
p22(x, t) =∏
x<Ti≤t,Zi<Ti
[1− 1
∑nj=1 I(Zj < Ti ≤ Tj)
]
is the Aalen-Johansen estimator of the transition probabilityp22(x, t), which is an
efficient estimator under the Markov condition.
9
The procedure is repeated until a bootstrap resample{(Z∗i , T
∗i )}n1
i=1 of size
n1 =∑n
j=1 I(Zj < Tj) is obtained.
3 Simulation study
We carried out a small simulation study to investigate the performance of the local
testτ t for a fixed grid oft-points and for the global testDn = sup∣∣∣τ t
∣∣∣. Specif-
ically, we have simulated 500 Monte Carlo trials of two different models. Both
models were based in the accelerated failure time specificationlog(T − Z) =
f(Z) + ε for the individuals passing through state 2, where the errorε is indepen-
dent of the ‘covariate’Z, whilef(.) is the predictor. An alternative representation
is given byλs (t) = λ0
((t− s) e−f(s)
)e−f(s) where (recall)λs (t) is the condi-
tional hazard ofT givenZ = s, Z < T andλ0 stands for the hazard ofW = eε.
This formulation belongs to the proportional hazards family whenε follows a
extreme-value distribution (i.e.λ0 is constant). On the other hand, the Markov
condition holds if and only ifλs (t) is free ofs. TheZ was distributed as aU [0, 2]
random variable and the models were as follows:
Model 0 (Markovian).λ0 constant andf(.) ≡ 0
Model 1 (Non-Markovian, proportional hazards).λ0 constant andf(s) =
(s− 1)2
The traces ofτt for Model 1 along the intervalt ∈ [0.5, 3] for several censoring
degrees are displayed in Figure 2. From this Figure 2 we can see that Model 1
presents first a negative and then a positive future-past association, and thatτt
vanish fort ≈ 1.8. This is a consequence of the increasing-decreasing shape of
the conditional hazardλs (t) = exp [−(s− 1)2] on the interval[0, 2] (note that
larger hazard values correspond to smallerT ’s). Of course, we omit the trace
corresponding to Model 0 because in this caseτt = 0 for eacht-value.
10
0.5 1.0 1.5 2.0 2.5
−0
.2−
0.1
0.0
0.1
0.2
tτ t
τt
τ~t, 10 % censτ~t, 30 % censτ~t, 50 % cens
Figure 2: Model 1. Comparison betweenτt and τt for several censoring rates in the
non-markovian simulated model. Monte Carlo approximation based on 100,000
observations.
The performance ofτ t was evaluated along a grid oft-values in the interval
[0.5,3]. Specifically, we tookt = 0.7, 1.1, 1.5, 1.9, 2.3, 2.7. The performance
of Dn = sup |τ t| was calculated over the same interval [0.5,3]. Two different
situations in terms of direct transitions between state 1 and state 3 (Z = T ) were
considered. Firstly, we assumed a situation with no direct transitions (the partic-
ular 3-state progressive model) and a more general situation where50% of the
individuals were assumed to transit directly from state 1 to state 3. Furthermore,
different sample sizes and censoring percentages were considered. As sample
sizes we tookn = 100, 250, and500. For the censoring distributionG, we took a
Uniform with support[0, bG], wherebG was chosen to obtain the desired censoring
level. We considered four situations from the uncensored case to a maximum of
50% of censored observations.
In Tables 1 to 4 we report the rejection proportion of the local test (when
based on the LB and the GB) along a grid oft-values, and for the correspond-
ing global test (based on the GB) for Models 0 and 1. A significance level of
5% was assumed, and we tookB = 200 bootstrap resamples. For comparison
purposes, the rejection proportion corresponding to the simple method based on
11
a proportional hazard specification with linear predictor,λs (t) = λ0(t)eβs, was
also included. It is interesting to see how the power of this test may not increase
with the sample size, as it happens in Tables 3 and 4 (Model 1). In this case, the
reason is found in the miss-specification of the predictor, which can not detect the
parabolic influencef(s) = (s − 1)2. A bit surprisingly, more power is reached
with larger censoring proportions; this is due to the shortening in the observable
support of theZ variable, which transforms the decreasing-increasing shape of
the predictor into a fairly monotone decreasing curve.
Note that Model 0 is Markovian, so we expect rejection proportions about
0.05 in this case. Results in Tables 1 (no direct transitions between state 1 and
state 3) and 2 (50% of direct transitions between state 1 and state 3) are quite
satisfactory to this regard. The nominal level is well approximated for both boot-
strap approaches in the local test. However, we observe that the the GB reports
more conservative results when increasing the censoring level. The global test
also presents satisfactory results, reporting rejection proportions of about5%.
For Model 1 we expect a rejection proportion increasing with the sample size
and also with the proportion of uncensoring. These features are appreciated in
Tables 3 (no direct transitions between state 1 and state 3) and 4 (50% of direct
transitions between state 1 and state 3). Interestingly, we see that the maximum
power of the local test based onτ t is achieved at some central point in the grid
(t = 1.1 in Model 1). This is a consequence of two facts. First, the amount
of information onτt grows at central points; second, we have that the largest
absolute value ofτt (see the black lines in Figure 2), and hence the alternative
most separated from the null, is obtained precisely fort = 1.1 (Model 1). It is
also remarkable the loss of power of the Kendall’s tau in Model 1 fort = 1.9; this
is because the level of association is zero at this point. So in practice the value of
t may have a big impact in the power of the test.
As for the global test , it is seen from tables 3 and 4 that its power may be
much larger than that of the proportional hazards approach, particularly for low to
12
moderate censoring degrees. For example, withn = 500 and 30 % of censoring,
the power of the nonparametric global test was more than four times (Table 3) or
three times (Table 4) that of the proportional hazards method. However, we must
indicate the loss of power observed when censoring level reaches the 50%.
Interestingly, if we compare Tables 3 and 4, we can observe a loss of power of
the new methods when the number of individuals passing through the intermediate
state declines. This is due to the reduction in the effective sample size to compute
τt as long as we enlarge the number of direct transitions1→3. While in the three-
state progressive model, where the direct transitions to state 3 without passing
through state 2 are not allowed, all the observations are used to compute the test
(Table 3), if we assume a 50% of direct transitions1→3 (Table 4) the effective
sample size is reduced to about half of the original sample size. So in practice,
the proportion of observed transitions through state 2 will dramatically affect the
performing of the new methods.
13
Table 1: Model 0 (Markovian). No transitions1→3. Rejection proportions of
tests based onτ t for several values oft andDn = sup |(ˆτ t|) along 500 trials,
censoring rates and sample sizes (n). Results corresponding to the proportional
hazard specification are also provided (PH method)% of censoring n Bootstrap method 0.7 1.1 1.5 1.9 2.3 2.7 Dn PH method
100 LB 0.042 0.050 0.028 0.062 0.040 0.062
GB 0.050 0.054 0.036 0.064 0.038 0.054 0.016 0.053
0 250 LB 0.040 0.049 0.045 0.046 0.043 0.051
GB 0.056 0.060 0.044 0.060 0.042 0.062 0.040 0.048
500 LB 0.045 0.050 0.042 0.068 0.046 0.040
GB 0.046 0.064 0.058 0.076 0.044 0.042 0.044 0.060
100 LB 0.048 0.052 0.036 0.046 0.050 0.044
GB 0.052 0.042 0.030 0.052 0.042 0.024 0.028 0.053
10 (bG = 20) 250 LB 0.049 0.056 0.048 0.053 0.045 0.044
GB 0.044 0.052 0.042 0.050 0.036 0.046 0.058 0.045
500 LB 0.041 0.042 0.049 0.063 0.053 0.042
GB 0.034 0.044 0.044 0.070 0.040 0.028 0.042 0.056
100 LB 0.044 0.046 0.044 0.064 0.068 0.062
GB 0.020 0.026 0.028 0.042 0.052 0.020 0.032 0.053
30 (bG = 6.6) 250 LB 0.045 0.046 0.049 0.058 0.049 0.053
GB 0.026 0.028 0.030 0.052 0.034 0.056 0.066 0.058
500 LB 0.053 0.055 0.045 0.060 0.042 0.045
GB 0.030 0.032 0.040 0.038 0.030 0.024 0.050 0.058
100 LB 0.020 0.022 0.016 0.030 0.014 0.076
GB 0.016 0.016 0.018 0.022 0.018 0.002 0.030 0.050
50 (bG = 3.9) 250 LB 0.043 0.042 0.041 0.055 0.049 0.043
GB 0.014 0.020 0.020 0.020 0.022 0.026 0.080 0.048
500 LB 0.061 0.045 0.060 0.052 0.043 0.049
GB 0.016 0.020 0.022 0.030 0.016 0.030 0.064 0.047
14
Table 2: Model 0 (Markovian). 50% transitions1→3. Rejection proportions of
tests based onτ t for several values oft andDn = sup |(ˆτ t|) along 500 trials,
censoring rates and sample sizes (n). Results corresponding to the proportional
hazard specification are also provided (PH method)% of censoring n Bootstrap method 0.7 1.1 1.5 1.9 2.3 2.7 Dn PH method
100 LB 0.064 0.064 0.054 0.062 0.066 0.084
GB 0.054 0.062 0.060 0.060 0.066 0.046 0.078 0.053
0 250 LB 0.046 0.046 0.034 0.048 0.055 0.042
GB 0.048 0.036 0.048 0.076 0.054 0.066 0.052 0.059
500 LB 0.043 0.043 0.051 0.062 0.054 0.070
GB 0.038 0.062 0.036 0.068 0.040 0.012 0.046 0.057
100 LB 0.048 0.058 0.042 0.054 0.066 0.074
GB 0.036 0.062 0.046 0.062 0.046 0.030 0.092 0.053
8 (bG = 20) 250 LB 0.051 0.048 0.049 0.049 0.044 0.049
GB 0.038 0.064 0.046 0.060 0.040 0.040 0.048 0.062
500 LB 0.063 0.047 0.050 0.049 0.051 0.068
GB 0.056 0.056 0.040 0.060 0.052 0.078 0.042 0.063
100 LB 0.052 0.064 0.058 0.064 0.082 -
GB 0.056 0.058 0.044 0.060 0.054 0.022 0.064 0.053
23 (bG = 6.6) 250 LB 0.051 0.066 0.053 0.058 0.052 0.057
GB 0.052 0.068 0.050 0.054 0.056 0.052 0.026 0.067
500 LB 0.050 0.049 0.060 0.061 0.046 0.069
GB 0.042 0.046 0.068 0.066 0.046 0.058 0.046 0.049
100 LB 0.036 0.038 0.034 0.022 0.070 -
GB 0.052 0.070 0.058 0.048 0.034 0.040 0.126 0.050
36 (bG = 3.9) 250 LB 0.050 0.057 0.046 0.049 0.050 -
GB 0.044 0.064 0.034 0.044 0.042 0.026 0.066 0.060
500 LB 0.052 0.055 0.053 0.045 0.055 -
GB 0.040 0.056 0.052 0.056 0.032 0.036 0.042 0.054
15
Table 3: Model 1 (Non-Markovian). No transitions1→3. Rejection proportions
of tests based onτ t for several values oft andDn = sup |(ˆτ t|) along 500 trials,
censoring rates and sample sizes (n). Results corresponding to the proportional
hazard specification are also provided (PH method)% of censoring n Bootstrap method 0.7 1.1 1.5 1.9 2.3 2.7 Dn PH method
100 LB 0.142 0.165 0.120 0.062 0.074 0.090
GB 0.252 0.280 0.158 0.084 0.088 0.084 0.230 0.082
0 250 LB 0.448 0.610 0.369 0.082 0.177 0.144
GB 0.460 0.610 0.390 0.088 0.164 0.140 0.600 0.088
500 LB 0.733 0.886 0.622 0.096 0.294 0.245
GB 0.736 0.884 0.616 0.116 0.292 0.258 0.882 0.087
100 LB 0.136 0.136 0.112 0.042 0.066 0.070
GB 0.178 0.224 0.138 0.058 0.088 0.066 0.190 0.072
10 (bG = 25) 250 LB 0.393 0.541 0.338 0.079 0.149 0.124
GB 0.384 0.536 0.324 0.076 0.132 0.106 0.518 0.060
500 LB 0.665 0.845 0.526 0.092 0.238 0.189
GB 0.652 0.808 0.506 0.104 0.248 0.202 0.834 0.098
100 LB 0.114 0.098 0.072 0.048 0.060 0.076
GB 0.086 0.162 0.082 0.042 0.056 0.044 0.104 0.072
30 (bG = 8.1) 250 LB 0.277 0.403 0.222 0.068 0.099 0.086
GB 0.228 0.322 0.178 0.056 0.074 0.060 0.298 0.073
500 LB 0.505 0.680 0.380 0.079 0.165 0.117
GB 0.442 0.618 0.368 0.068 0.146 0.100 0.482 0.118
100 LB 0.096 0.084 0.072 0.062 0.088 -
GB 0.062 0.088 0.056 0.046 0.030 0.016 0.048 0.070
50 (bG = 4.6) 250 LB 0.200 0.264 0.144 0.058 0.067 0.068
GB 0.088 0.170 0.078 0.050 0.028 0.038 0.068 0.108
500 LB 0.371 0.462 0.239 0.069 0.102 0.076
GB 0.264 0.356 0.180 0.052 0.058 0.046 0.176 0.172
16
Table 4: Model 1 (Non-Markovian). 50% transitions1→3. Rejection proportions
of tests based onτ t for several values oft andDn = sup |(ˆτ t|) along 500 trials,
censoring rates and sample sizes (n). Results corresponding to the proportional
hazard specification are also provided (PH method)% of censoring n Bootstrap method 0.7 1.1 1.5 1.9 2.3 2.7 Dn PH method
100 LB 0.142 0.160 0.120 0.062 0.074 0.090
GB 0.140 0.184 0.116 0.052 0.078 0.082 0.102 0.082
0 250 LB 0.218 0.324 0.210 0.083 0.146 0.128
GB 0.222 0.326 0.200 0.096 0.150 0.134 0.275 0.089
500 LB 0.454 0.585 0.380 0.088 0.177 0.146
GB 0.470 0.600 0.398 0.090 0.174 0.162 0.624 0.091
100 LB 0.136 0.136 0.112 0.042 0.066 0.070
GB 0.120 0.130 0.116 0.046 0.064 0.062 0.094 0.072
8 (bG = 25) 250 LB 0.202 0.284 0.164 0.081 0.123 0.110
GB 0.192 0.290 0.158 0.074 0.128 0.106 0.230 0.091
500 LB 0.400 0.525 0.339 0.089 0.155 0.124
GB 0.406 0.534 0.366 0.092 0.168 0.136 0.522 0.093
100 LB 0.114 0.098 0.072 0.048 0.060 0.076
GB 0.114 0.104 0.086 0.054 0.052 0.060 0.064 0.072
23 (bG = 8.1) 250 LB 0.154 0.192 0.127 0.063 0.101 0.089
GB 0.156 0.178 0.108 0.070 0.108 0.076 0.142 0.067
500 LB 0.290 0.385 0.219 0.081 0.116 0.090
GB 0.280 0.392 0.218 0.086 0.096 0.096 0.296 0.086
100 LB 0.096 0.084 0.072 0.062 0.055 -
GB 0.088 0.086 0.072 0.056 0.050 0.034 0.256 0.070
36 (bG = 4.6) 250 LB 0.094 0.143 0.080 0.046 0.071 0.068
GB 0.096 0.130 0.082 0.048 0.060 0.054 0.054 0.076
500 LB 0.197 0.259 0.124 0.067 0.080 0.076
GB 0.206 0.292 0.132 0.078 0.076 0.060 0.090 0.098
4 Real data illustration
Two datasets were analyzed in order to illustrate the new methods. Because of the
presence of ties in the real data, the discrete version ofτ t, τ t, was used:
τ t =τ t
pc,t + pd,t=pc,t − pd,tpc,t + pd,t
17
4.1 Bone marrow transplantation for leukemia
We analyzed survival data from a multicenter trial of patients with acute myeloctic
leukemia after a bone marrow transplantation. The details of this dataset can be
found in [9]. The recovering from a bone marrow transplant is a complex process
in patients with leukemia and their prognosis may be affected by intermediate
events, such as the development of chronic graft-versus-host disease (cGVHD).
Therefore, we considered an illness-model to analyze the prognostic of the pa-
tients after the bone marrow transplant, considering the development of cGVHD
as intermediate event of interest.
Specifically, we studied the recovery process of 136 patients. 60 of them de-
veloped cGHVD after the transplant (18% of censoring in state 1) and among
them, 28 died (24% of censoring in state 2). Furthermore, 52 patients (38%) were
observed to die without developing cGHVD. We studied the markovianity in the
range of observed values from 150 to 500 days of follow-up time, so that we guar-
antee a minimum of 35 observations to perform eachτ t. In Figure 3 we report
two graphics to summarize the application of the local markovianity test to this
dataset. In the left panel we give the trace of values ofτ t, with its corresponding
95% and 90% pointwise confidence bands based on the simple bootstrap. This
Figure suggests a significant positive future-past association in the whole range
of time, that should be interpreted as an increased risk of death for those patients
suffering cGVHD shortly after the bone marrow transplant. In the right panel,
we report the significance trace of the local goodness-of-fit based onτ t. Results
for both local and global bootstrap methods introduced in section 2.1 are shown,
and they basically report the same trace. We observe that the significance trace
is able to detect non-markovianity along all the range of t-values. Accordingly,
when we apply the summary test based on the supremum over the observations
falling in the interval of interest, a p-value=0.014 was obtained. Interestingly, this
agrees with the analysis of markovianity based on the proportional hazards model
λs(t) = λ0(t)eβs under whichβ = −0.005 (s.e. = 0.00203) with a logrank test’s
18
p-value of 0.011.
150 200 250 300 350 400 450 500
−1
.0−
0.5
0.0
0.5
1.0
time (days)
τ t
150 200 250 300 350 400 450 500
0.0
0.2
0.4
0.6
0.8
1.0
time (days)
p−
valu
e
Figure 3: Left: Future-past association trace for the leukemia data and pointwise
confidence limits. Solid line: 95%. Dashed line: 90%. Right: Significance trace.
B=2000 bootstrap resamples. Solid line: GB. Dashed line: LB
4.2 Infection for burn patients
The second dataset refers to the study of time from admission until either infec-
tion with Staphylococcus aureus or discharge in patients from the burn unit at a
large university hospital ([10]). The infection of a burn wound is a common com-
plication for these patients that might be affected for other factors, such as days
until first wound excision during the evolution process. Both, the time to Straphy-
locous Aureaus (T ) infection and time to excision (Z) are considered in this case,
and hence an illness-death model applies. From a total of 154 patients studied,
the patient’s wound had been excised for 84 (23% of censoring in state 1). From
them, 14 patients experienced infection (83% of censoring in state 2). Besides, 34
patients experienced the infection without having passed through the intermediate
event.
In this case, the Cox method accepts the global markovianity at 5% signifi-
19
cance level, providing a large p-value (p-value=0.448). However, when applying
our methods, the results does not show that clear markovian performance. We
studied the local markovianity along the interval [7,31] days from the beginning
of the follow-up time to guarantee, as in the previous case, 35 observations in each
correspondingAt. In Figure 4 (left panel), we can observe the estimated future-
past association for the burn data. It suggests a negative association between the
time to excision and the total time at the t-interval [14,20] days of follow-up time,
meaning that the risk of Straphylocous Aureaus is increased after the excision.
This is confirmed by the significance trace of the local test (Figure 4 right). The
local method rejects the local markovianity at a 5% of significance level at similar
part of the grid of t-points. As for the global test, we obtain p-value=0.144. Even
if the global test does not reach the significance level, the p-value associated is
quite lower than the one provided by the classical proportional hazard method.
20
10 15 20 25 30
−1
.0−
0.5
0.0
0.5
1.0
time (days)
τ t
10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
time (days)
p−
valu
e
Figure 4: Left: Future-past association trace for the burn data and pointwise con-
fidence limits. Solid line: 95%. Dashed line: 90%. Right: Significance trace.
B=2000 bootstrap resamples. Solid line: GB. Dashed line: LB
5 Conclusions
In this paper, a flexible nonparametric method to test the Markov condition in the
scope of the illness-death model has been proposed. The nonparametric test is
based on the measured association between the past and the future of the process
of interest given its present state at time t. By considering several values oft, one
may construct a significance trace which reports information on the markovianity
of the process in a local way. Besides, we have proposed a supremum-type test
statistic by considering the maximum observed absolute future-past association
over a given time interval. The weak convergence of the underlying test statistic
has been established, and several bootstrap approximations have been proposed.
The obtained simulation results suggest that the new test may be much more
powerful than existing, less flexible methods, for special alternatives. This has
been illustrated through real medical data analysis too. Specifically, we have ana-
lyzed the impact of chronic graft-versus-disease in evolution after a bone marrow
21
transplant in patients with leukemia, and also the relationship between the time to
wound excision and time to Straphylocous Aureaus infection in burn patients. In
both cases, the Markov condition for the corresponding illness-death model was
tested, and the new method gave new interesting insights into the problem.
The relevance of the proposed test comes from the fact that no special depen-
dence structure between the future and the past of the process is a priori assumed.
Interestingly, a characterization of the Markov condition in terms of the marko-
vianity of the censored, observable process has been given, in such a way that
no Kaplan-Meier weights are needed for the computation of the proposed testing
algorithms. Although a Kaplan-Meier-based formula for the association measure
is possible, it is expected that it would lead to a less powerful test due to the large
variance which would be typically obtained in heavily censored scenarios.
It would be interesting to extend the given methods to other more involved
multi-state models. The extension to other multi-state models can become com-
plicated as the number of states and possible transition grow. However, the method
could be applied as it is to investigate markovianity in specific parts of a general
multi-state model, under the assumption that the only source of possible non-
markovianity is the sojourn time in the previously visited state.
All the proposed methods were implemented in R. The code is available from
the authors upon request.
Acknowledgements
Work supported by the Grants MTM2008-03129 of the Spanish Ministerio de
Ciencia e Innovacion and 10PXIB300068PRof the Xunta de Galicia. Financial
support from the INBIOMED project (DXPCTSUG, Ref. 2009/063) is also ac-
knowledged.
22
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24
Appendix: Technical proofs
Proof to Theorem 1.We begin stating two useful Lemmas. LetbH = inf {t : H(t) = 1},
whereH is the df ofT . PutF1t(x) = Ft(x,∞) andF2t(y) = Ft(∞, y) = Ft(t, y)
for the marginal distributions ofZ andT respectively givenAt. When needed, we
also use the notationsF1t(x) = Ft(x,∞) = Ft(x, bH) andF2t(y) = Ft(∞, y) =
Ft(t, y) for the marginal distributions ofZ andT respectively givenAt.
Lemma A. We have for allx ≤ t < y ≤ bH
Ft(x, y) = F1t(x)− P (C > y|C > t) [F1t(x)− Ft(x, y)] .
Proof. Write for x ≤ t < y ≤ bH
Ft(x, y) = P (Z ≤ x, T ≤ y|At) = P (Z ≤ x, T ∧ C ≤ y|Z ≤ t < T, C > t)
= P (Z ≤ x|Z ≤ t < T, C > t)− P (Z ≤ x, T ∧ C > y|Z ≤ t < T, C > t)
= F1t(x)− P (Z ≤ x, T > y|Z ≤ t < T )P (C > y|C > t)
= F1t(x)− P (C > y|C > t) [F1t(x)− Ft(x, y)] .�
Corollary. We haveF1t(x) = F1t(x) for all x ≤ t, and
F2t(y) = 1− P (C > y|C > t) [1− F2t(y)] for all y ≤ bH .
Proof. For the first assertion note that, by Lemma A,
F1t(x) = Ft(x, bH) = F1t(x)−P (C > bH |C > t) [F1t(x)− Ft(x, bH)] = F1t(x),
where the last equality follows becauseP (C > bH |C > t) = 0 whenever
Ft(x, bH) < F1t(x). The second assertion follows directly from Lemma A.�
Lemma B. The two following conditions are equivalent:
(i) Ft(x, y) = F1t(x)F2t(y) for all x ≤ t < y ≤ bH (i.e. Z and T are
independent givenAt)
25
(ii) Ft(x, y) = F1t(x)F2t(y) for all x ≤ t < y ≤ bH
Proof. Assume that (i) holds. Then, by Lemma A and its Corollary, we have
F1t(x)− P (C > y|C > t) [F1t(x)− Ft(x, y)] =
= F1t(x)− F1t(x)P (C > y|C > t) [1− F2t(y)] ,
which holds only if
P (C > y|C > t)Ft(x, y) = P (C > y|C > t)F1t(x)F2t(y).
But this is just (ii), after noting thatP (C > y|C > t) > 0 for y < bH . Conversely,
if (ii) holds, then (i) inmediately follows from Lemma A and its Corollary.�
Lemma B implies the two assertions of Theorem 2. Note that, when the sup-
port ofT is contained in that ofC, we have that (ii) holds if and only ifZ andT
are conditionally independent givenAt.�
Proof to Theorem 2:
We want to prove that the process{√
n(τ t − τt) : t ∈ I}
is tight, wereIstands for an interval such thatP (At) ≥ c > 0 for all t ∈ I. Recall that
τ t = pc,t − pd,t
where
pc,t = 2
∫ ∫F t(x
−, y−)F t(dx, dy) and pd,t = 2
∫ ∫U t(x
−, y−)F t(dx, dy)
and whereF t is the ordinary empirical df of the(Zi, Ti
)’s such thatZi ≤ t < Ti
and U t(x, y) =F t(t, y
−) − F t(x
−, y−). We first obtain an asymptotic represen-
tation of{√
n(τ t − τt) : t ∈ I}
as a suitable process plus a remainder.
It is easily seen that
pc,t =2
nt(nt − 1)
∑
i<j
I((Zi − Zj)(Ti − Tj) > 0)I(Zi ≤ t < Ti)I(Zj ≤ t < Tj)
26
wherent =∑n
i=1 I(Zi ≤ t < Ti). Sincent/n → P (At) asn → ∞, we obtain
pc,t =2
n(n− 1)P (At)2
∑
i<j
I((Zi−Zj)(Ti−Tj) > 0)I(Zi ≤ t < Ti)I(Zj ≤ t < Tj)+oP (1)
uniformly ont ∈ I. A similar result holds forpd,t and thus we have
τ t =2
n(n− 1)P (At)2
∑
i6=j
ϕt(Zi, Zj, Ti, Tj)− 1 + oP (1)
where
ϕt(Zi, Zj, Ti, Tj) = I((Zi − Zj)(Ti − Tj) > 0)I(Zi ≤ t < Ti)I(Zj ≤ t < Tj).
Now, straightforward calculations show that the Hajek projection of theU-statistic
Ut =1
n(n− 1)
∑
i6=j
ϕt(Zi, Zj, Ti, Tj)
is given by (cfr. [17], Section 5.3)
Ut =2P (At)
n
n∑
i=1
[St(Zi, Ti) + Ft(Zi, Ti)
]I(Zi ≤ t < Ti)− P (At)
2pc,t
whereSt(x, y) = P (Z > x, T > y|At). SinceUt − Ut = oP (n−1/2) we have
√n(τ t − τt) =
√n
[2
P (At)2Ut − 1− (2pc,t − 1)
]+ oP (1)
=√n
[2
P (At)2(Ut − P (At)
2pc,t)
]+ oP (1)
= n−1/2 4
P (At)
n∑
i=1
{[St(Zi, Ti) + Ft(Zi, Ti)
]I(Zi ≤ t < Ti)− P (At)pc,t
}+ oP (1)
uniformly on t ∈ I. Hence, tightness of√n(τ t − τt) follows provided that the
process
Rn(t) = n−1/2n∑
i=1
{[St(Zi, Ti) + Ft(Zi, Ti)
]I(Zi ≤ t < Ti)− P (At)pc,t
}
27
is tight. This is stated as a Lemma, which completes the proof.�
Lemma. The process{Rn(t) : t ∈ I} is tight.
Proof to Lemma. For t1 ≤ t ≤ t2 write
E{|Rn(t)− Rn(t1)|2 |Rn(t2)− Rn(t)|2
}= E
(n−1/2
n∑
i=1
αi
)2(n−1/2
n∑
i=1
βi
)2
where
αi =[St(Zi, Ti) + Ft(Zi, Ti)
]I(Zi ≤ t < Ti)− P (At)pc,t
−[St1(Zi, Ti) + Ft1(Zi, Ti)
]I(Zi ≤ t1 < Ti) + P (At1)pc,t1
and
βi =[St2(Zi, Ti) + Ft2(Zi, Ti)
]I(Zi ≤ t2 < Ti)− P (At2)pc,t2
−[St(Zi, Ti) + Ft(Zi, Ti)
]I(Zi ≤ t < Ti) + P (At)pc,t,
1 ≤ i ≤ n. Note thatE(αi) = E(βi) = 0. Use a symmetry argument to write
E
(n−1/2
n∑
i=1
αi
)2(n−1/2
n∑
i=1
βi
)2
= n−2{nE(α2
1β21) + n(n− 1)E(α2
1)E(β21) + 2n(n− 1)E2(α1β1)
}
≤ n−2{nE(α2
1β21) + 3n(n− 1)E(α2
1)E(β21)},
the inequality following from Cauchy-Schwarz. Now, usingZi ≤ Ti andt1 ≤ t
we get
I(Zi ≤ t1 < Ti) = I(Ti > t1)− I(Zi > t1) (1)
= I(Ti > t) + I(t1 < Ti ≤ t)− I(Zi > t)− I(t1 < Zi ≤ t)
= I(Zi ≤ t < Ti) + I(t1 < Ti ≤ t)− I(t1 < Zi ≤ t).
Hence,
αi =[St(Zi, Ti)− St1(Zi, Ti)
]I(Zi ≤ t < Ti)
28
+[Ft(Zi, Ti)− Ft1(Zi, Ti)
]I(Zi ≤ t < Ti)
−[St1(Zi, Ti) + Ft1(Zi, Ti)
]I(t1 < Ti ≤ t)+
[St1(Zi, Ti) + Ft1(Zi, Ti)
]I(t1 < Zi ≤ t)
−P (At)pc,t + P (At1)pc,t1,
while for the last term, using again (1), we obtain
−P (At)pc,t+P (At1)pc,t1 = P (At) [pc,t1 − pc,t]+pc,t1P (t1 < T ≤ t)−pc,tI(t1 < Z ≤ t).
Now, noting thatpc,t = 12E[Ft(Z, T )I(Z ≤ t < T )
]/P (At), we have
2(pc,t1 − pc,t) =1
P (At1)E[Ft1(Z, T )(I(Z ≤ t < T ) + I(t1 < T ≤ t)− I(t1 < Z ≤ t))
]
− 1
P (At1)E[Ft(Z, T )I(Z ≤ t < T )
]
−(1
P (At)− 1
P (At1))E[Ft(Z, T )I(Z ≤ t < T )
]
=1
P (At1)E[(Ft1(Z, T )− Ft(Z, T ))I(Z ≤ t < T )
]
+1
P (At1)E[Ft1(Z, T )(I(t1 < T ≤ t)− I(t1 < Z ≤ t))
]
−P (At1)− P (At)
P (At)P (At1)E[Ft(Z, T )I(Z ≤ t < T )
].
Since (1) implies for some constantk > 0
∣∣∣St(Zi, Ti)− St1(Zi, Ti)∣∣∣ ≤ k
[P (t1 < T ≤ t) + P (t1 < Z ≤ t)
]= k [Γ(t)− Γ(t1)]
and ∣∣∣Ft(Zi, Ti)− Ft1(Zi, Ti)∣∣∣ ≤ k [Γ(t)− Γ(t1)]
whereΓ(t) = P (T ≤ t) + P (Z ≤ t1) is a continuous, nondecreasing function,
and sinceFt, St ∈ [0, 1], from (a + b)2 ≤ 2(a2 + b2) we have for some constant
k′ > 0
E(α2i ) ≤ k′
{[Γ(t)− Γ(t1)]
2 + Γ(t)− Γ(t1)}.
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An analogous result can be proved forβi and hence
E(α21)E(β2
1) ≤ k′′{[Γ(t2)− Γ(t1)]
4 + [Γ(t2)− Γ(t1)]2} .
Similarly, one may obtainE(α21β
21) ≤ k′′′ [Γ(t2)− Γ(t1)]
2 and the result follows
from Theorem 15.6 in [18].�
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